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Theorem imasf1oxmet 24406

Description: The image of an extended metric is an extended metric. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Hypotheses**
Ref	Expression
imasf1oxmet.u	⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
imasf1oxmet.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
imasf1oxmet.f	⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
imasf1oxmet.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)
imasf1oxmet.e	⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
imasf1oxmet.d	⊢ 𝐷 = (dist‘𝑈)
imasf1oxmet.m	⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))

**Assertion**
Ref	Expression
imasf1oxmet	⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))

Proof of Theorem imasf1oxmet Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasf1oxmet.u	. . . 4 ⊢ (𝜑 → 𝑈 = (𝐹 “_s 𝑅))
2		imasf1oxmet.v	. . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑅))
3		imasf1oxmet.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)
4		f1ofo 6869	. . . . 5 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–onto→𝐵)
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵)
6		imasf1oxmet.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑍)
7		eqid 2740	. . . 4 ⊢ (dist‘𝑅) = (dist‘𝑅)
8		imasf1oxmet.d	. . . 4 ⊢ 𝐷 = (dist‘𝑈)
9	1, 2, 5, 6, 7, 8	imasdsfn 17574	. . 3 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵))
10	1	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑈 = (𝐹 “_s 𝑅))
11	2	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑉 = (Base‘𝑅))
12	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐹:𝑉–1-1-onto→𝐵)
13	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑅 ∈ 𝑍)
14		imasf1oxmet.e	. . . . . . . 8 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))
15		imasf1oxmet.m	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))
16	15	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐸 ∈ (∞Met‘𝑉))
17		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ 𝑉)
18		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉)
19	10, 11, 12, 13, 14, 8, 16, 17, 18	imasdsf1o 24405	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = (𝑎𝐸𝑏))
20		xmetcl 24362	. . . . . . . . 9 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎𝐸𝑏) ∈ ℝ^*)
21	20	3expb 1120	. . . . . . . 8 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ^*)
22	15, 21	sylan 579	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ^*)
23	19, 22	eqeltrd 2844	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*)
24	23	ralrimivva 3208	. . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*)
25		f1ofn 6863	. . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹 Fn 𝑉)
26	3, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉)
27		oveq2 7456	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎)𝐷𝑦) = ((𝐹‘𝑎)𝐷(𝐹‘𝑏)))
28	27	eleq1d 2829	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
29	28	ralrn 7122	. . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
30	26, 29	syl 17	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^*))
31		forn 6837	. . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵)
32	5, 31	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵)
33	32	raleqdv 3334	. . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
34	30, 33	bitr3d 281	. . . . . 6 ⊢ (𝜑 → (∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
35	34	ralbidv 3184	. . . . 5 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
36	24, 35	mpbid 232	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*)
37		oveq1 7455	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥𝐷𝑦) = ((𝐹‘𝑎)𝐷𝑦))
38	37	eleq1d 2829	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) ∈ ℝ^* ↔ ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
39	38	ralbidv 3184	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
40	39	ralrn 7122	. . . . . 6 ⊢ (𝐹 Fn 𝑉 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
41	26, 40	syl 17	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^*))
42	32	raleqdv 3334	. . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^* ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
43	41, 42	bitr3d 281	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ^* ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
44	36, 43	mpbid 232	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*)
45		ffnov 7576	. . 3 ⊢ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ↔ (𝐷 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ^*))
46	9, 44, 45	sylanbrc 582	. 2 ⊢ (𝜑 → 𝐷:(𝐵 × 𝐵)⟶ℝ^*)
47		xmeteq0 24369	. . . . . . . 8 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → ((𝑎𝐸𝑏) = 0 ↔ 𝑎 = 𝑏))
48	16, 17, 18, 47	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝑎𝐸𝑏) = 0 ↔ 𝑎 = 𝑏))
49	19	eqeq1d 2742	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝑎𝐸𝑏) = 0))
50		f1of1 6861	. . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–1-1→𝐵)
51	3, 50	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑉–1-1→𝐵)
52		f1fveq 7299	. . . . . . . 8 ⊢ ((𝐹:𝑉–1-1→𝐵 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏))
53	51, 52	sylan 579	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏))
54	48, 49, 53	3bitr4d 311	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
55	16	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝐸 ∈ (∞Met‘𝑉))
56		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑐 ∈ 𝑉)
57	17	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑎 ∈ 𝑉)
58	18	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑏 ∈ 𝑉)
59		xmettri2 24371	. . . . . . . . . 10 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑐 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ≤ ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
60	55, 56, 57, 58, 59	syl13anc 1372	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → (𝑎𝐸𝑏) ≤ ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
61	19	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = (𝑎𝐸𝑏))
62	10	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑈 = (𝐹 “_s 𝑅))
63	11	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑉 = (Base‘𝑅))
64	12	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝐹:𝑉–1-1-onto→𝐵)
65	13	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → 𝑅 ∈ 𝑍)
66	62, 63, 64, 65, 14, 8, 55, 56, 57	imasdsf1o 24405	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑐)𝐷(𝐹‘𝑎)) = (𝑐𝐸𝑎))
67	62, 63, 64, 65, 14, 8, 55, 56, 58	imasdsf1o 24405	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑐)𝐷(𝐹‘𝑏)) = (𝑐𝐸𝑏))
68	66, 67	oveq12d 7466	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) = ((𝑐𝐸𝑎) +_𝑒 (𝑐𝐸𝑏)))
69	60, 61, 68	3brtr4d 5198	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ 𝑐 ∈ 𝑉) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
70	69	ralrimiva 3152	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
71		oveq1 7455	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑐) → (𝑧𝐷(𝐹‘𝑎)) = ((𝐹‘𝑐)𝐷(𝐹‘𝑎)))
72		oveq1 7455	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝐹‘𝑐) → (𝑧𝐷(𝐹‘𝑏)) = ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))
73	71, 72	oveq12d 7466	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘𝑐) → ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) = (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))))
74	73	breq2d 5178	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘𝑐) → (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
75	74	ralrn 7122	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝑉 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
76	26, 75	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏)))))
77	32	raleqdv 3334	. . . . . . . . 9 ⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
78	76, 77	bitr3d 281	. . . . . . . 8 ⊢ (𝜑 → (∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
79	78	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (∀𝑐 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ (((𝐹‘𝑐)𝐷(𝐹‘𝑎)) +_𝑒 ((𝐹‘𝑐)𝐷(𝐹‘𝑏))) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
80	70, 79	mpbid 232	. . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))
81	54, 80	jca 511	. . . . 5 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
82	81	ralrimivva 3208	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
83	27	eqeq1d 2742	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) = 0 ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0))
84		eqeq2 2752	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎) = 𝑦 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)))
85	83, 84	bibi12d 345	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ↔ (((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏))))
86		oveq2 7456	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝐹‘𝑏) → (𝑧𝐷𝑦) = (𝑧𝐷(𝐹‘𝑏)))
87	86	oveq2d 7464	. . . . . . . . . . 11 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))
88	27, 87	breq12d 5179	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
89	88	ralbidv 3184	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))))
90	85, 89	anbi12d 631	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑏) → (((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
91	90	ralrn 7122	. . . . . . 7 ⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
92	26, 91	syl 17	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏))))))
93	32	raleqdv 3334	. . . . . 6 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
94	92, 93	bitr3d 281	. . . . 5 ⊢ (𝜑 → (∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
95	94	ralbidv 3184	. . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = 0 ↔ (𝐹‘𝑎) = (𝐹‘𝑏)) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷(𝐹‘𝑏)))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
96	82, 95	mpbid 232	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
97	37	eqeq1d 2742	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) = 0 ↔ ((𝐹‘𝑎)𝐷𝑦) = 0))
98		eqeq1 2744	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥 = 𝑦 ↔ (𝐹‘𝑎) = 𝑦))
99	97, 98	bibi12d 345	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ↔ (((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦)))
100		oveq2 7456	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑧𝐷𝑥) = (𝑧𝐷(𝐹‘𝑎)))
101	100	oveq1d 7463	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) = ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))
102	37, 101	breq12d 5179	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) ↔ ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
103	102	ralbidv 3184	. . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)) ↔ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))))
104	99, 103	anbi12d 631	. . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → ((((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
105	104	ralbidv 3184	. . . . . 6 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
106	105	ralrn 7122	. . . . 5 ⊢ (𝐹 Fn 𝑉 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
107	26, 106	syl 17	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦)))))
108	32	raleqdv 3334	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))))
109	107, 108	bitr3d 281	. . 3 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((((𝐹‘𝑎)𝐷𝑦) = 0 ↔ (𝐹‘𝑎) = 𝑦) ∧ ∀𝑧 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ≤ ((𝑧𝐷(𝐹‘𝑎)) +_𝑒 (𝑧𝐷𝑦))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦)))))
110	96, 109	mpbid 232	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))
111	15	elfvexd 6959	. . . 4 ⊢ (𝜑 → 𝑉 ∈ V)
112		focdmex 7996	. . . 4 ⊢ (𝑉 ∈ V → (𝐹:𝑉–onto→𝐵 → 𝐵 ∈ V))
113	111, 5, 112	sylc 65	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
114		isxmet 24355	. . 3 ⊢ (𝐵 ∈ V → (𝐷 ∈ (∞Met‘𝐵) ↔ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
115	113, 114	syl 17	. 2 ⊢ (𝜑 → (𝐷 ∈ (∞Met‘𝐵) ↔ (𝐷:(𝐵 × 𝐵)⟶ℝ^* ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (((𝑥𝐷𝑦) = 0 ↔ 𝑥 = 𝑦) ∧ ∀𝑧 ∈ 𝐵 (𝑥𝐷𝑦) ≤ ((𝑧𝐷𝑥) +_𝑒 (𝑧𝐷𝑦))))))
116	46, 110, 115	mpbir2and 712	1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 class class class wbr 5166 × cxp 5698 ran crn 5701 ↾ cres 5702 Fn wfn 6568 ⟶wf 6569 –1-1→wf1 6570 –onto→wfo 6571 –1-1-onto→wf1o 6572 ‘cfv 6573 (class class class)co 7448 0cc0 11184 ℝ^*cxr 11323 ≤ cle 11325 +_𝑒 cxad 13173 Basecbs 17258 distcds 17320 “_s cimas 17564 ∞Metcxmet 21372

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-0g 17501 df-gsum 17502 df-xrs 17562 df-imas 17568 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-xmet 21380

This theorem is referenced by: imasf1omet 24407 xpsxmet 24411 imasf1obl 24522 imasf1oxms 24523

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